Private graphon estimation via sum-of-squares
This provides a practical solution for privacy-preserving network analysis, addressing a bottleneck in applying differential privacy to graph data, though it is incremental in improving computational efficiency over prior work.
The paper tackles the problem of estimating graphons and stochastic block models with node-differential privacy, achieving polynomial-time algorithms that match the statistical utility of previous exponential-time private mechanisms for any constant number of blocks.
We develop the first pure node-differentially-private algorithms for learning stochastic block models and for graphon estimation with polynomial running time for any constant number of blocks. The statistical utility guarantees match those of the previous best information-theoretic (exponential-time) node-private mechanisms for these problems. The algorithm is based on an exponential mechanism for a score function defined in terms of a sum-of-squares relaxation whose level depends on the number of blocks. The key ingredients of our results are (1) a characterization of the distance between the block graphons in terms of a quadratic optimization over the polytope of doubly stochastic matrices, (2) a general sum-of-squares convergence result for polynomial optimization over arbitrary polytopes, and (3) a general approach to perform Lipschitz extensions of score functions as part of the sum-of-squares algorithmic paradigm.